Computational Complexity and other fun stuff in math and computer science from Lance Fortnow and Bill Gasarch

Tuesday, March 30, 2004

Changes in Introductory Theory

Comments to my last
post basically ask how has the introductory courses in theory has
changed over the years. My first reaction: remarkably
little. Theoretical models of computation do not depend on the current
hot technology, particularly at the undergraduate level. Many of the
basic results we teach today were also taught say 25 years ago. But
without doubt theory courses have changed their emphasis on various
topics over the years.

Every professor teaches a theory course differently so there is no
fixed answer to what has changed. But here are some trends that I have seen
(from a distinctly American point of view):

Less emphasis on automata theory, particularly for context-free
languages. Many schools do away with automata theory all together.

Less depth in computability theory. Most courses will cover
undecidability but you'll less often see the recursion theory or even
Rice's theorem taught.

Does anybody still teach the Gap, Union
and Speed-Up Theorems and Blum complexity measures anymore?

Only
one new theorem since the mid-70's has become a fundamental part of an
undergraduate complexity course: The 1988 Immerman-Szelepcsényi
Theorem stating that nondeterministic space is closed under
complement.

There has been a trend in adding some recent research
in complexity as the end of a course based on the interests of the
instructor: Randomized computation (though recent algorithms for
primality might change how it gets taught), cryptography, interactive
proofs, PCPs and approximation, quantum computing for
example. Parallel computation has come and gone.

But remember these are exceptions. Basic topics like Turing machines,
undecidability, NP-completeness, Savitch's theorem and time and space
hierarchies still get taught much the way they were taught in the 70's.